Ebook: Theory of Complex Functions

R. Remmert and R.B. Burckel

Theory of Complex Functions

"Its accessibility makes it very useful for a first graduate course on complex function theory, especially where there is an opportunity for developing an interest on the part of motivated students in the history of the subject. Historical remarks abound throughout the text. Short biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass are given. There is an extensive bibliography of classical works on complex function theory with comments on some of them. In addition, a list of modern complex function theory texts and books on the history of the subject and of mathematics is given. Throughout the book there are numerous interesting quotations. In brief, the book affords splendid opportunities for a rich treatment of the subject."—MATHEMATICAL REVIEWS

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The material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure.

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The material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure.
Content:
Front Matter....Pages i-xix
Historical Introduction....Pages 1-7
Complex Numbers and Continuous Functions....Pages 9-44
Complex-Differential Calculus....Pages 45-70
Holomorphy and Conformality. Biholomorphic Mappings....Pages 71-89
Modes of Convergence in Function Theory....Pages 91-107
Power Series....Pages 109-132
Elementary Transcendental Functions....Pages 133-165
Complex Integral Calculus....Pages 167-189
The Integral Theorem, Integral Formula and Power Series Development....Pages 191-225
Fundamental Theorems about Holomorphic Functions....Pages 227-263
Miscellany....Pages 265-301
Isolated Singularities. Meromorphic Functions....Pages 303-320
Convergent Series of Meromorphic Functions....Pages 321-341
Laurent Series and Fourier Series....Pages 343-375
The Residue Calculus....Pages 377-393
Definite Integrals and the Residue Calculus....Pages 395-415
Back Matter....Pages 417-453

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The material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure.
Content:
Front Matter....Pages i-xix
Historical Introduction....Pages 1-7
Complex Numbers and Continuous Functions....Pages 9-44
Complex-Differential Calculus....Pages 45-70
Holomorphy and Conformality. Biholomorphic Mappings....Pages 71-89
Modes of Convergence in Function Theory....Pages 91-107
Power Series....Pages 109-132
Elementary Transcendental Functions....Pages 133-165
Complex Integral Calculus....Pages 167-189
The Integral Theorem, Integral Formula and Power Series Development....Pages 191-225
Fundamental Theorems about Holomorphic Functions....Pages 227-263
Miscellany....Pages 265-301
Isolated Singularities. Meromorphic Functions....Pages 303-320
Convergent Series of Meromorphic Functions....Pages 321-341
Laurent Series and Fourier Series....Pages 343-375
The Residue Calculus....Pages 377-393
Definite Integrals and the Residue Calculus....Pages 395-415
Back Matter....Pages 417-453
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